Abstract
In this paper, we prove that if a, b and c are pairwise coprime positive integers such that a 2 + b 2 = c r , a > b, a ≡ 3 (mod 4), b ≡ 2 (mod 4) and c−1 is not a square, then a x + b y = c z has only the positive integer solution (x, y, z) = (2, 2, r). Let m and r be positive integers with 2|m and 2 Χr, define the integers U r , V r by (m + √−1) r = V r + U r √−1. If a = |U r |, b = |V r |, c = m 2 + 1 with m ≡ 2 (mod 4), a ≡ 3 (mod 4), and if r < m/√1.5 log3(m 2 + 1)−1, then a x + b y = c z has only the positive integer solution (x, y, z) = (2, 2, r). The argument here is elementary.
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